MISCELLANEOUS QUESTIONS, APPLICABLE TO THE VARIOUS HEADS OF THE PRECEDING VOLUME. N.B. Answers, worked at length, are given in THE KEY. 1.-A RECKONING of 20 shillings was spent by a company of twenty persons, consisting of officers, sailors, and marines. Each officer paid 2s. 6d., each sailor 12 pence, and cach marine 8 pence. How many persons were there of each denomination? 2.-A gentleman has a piece of land in the form of an isosceles triangle the perimeter of which is to be 144 chains, to be taken out of a large meadow: Now, his bargain is, to given ten guineas for every chain of the perpendicular, and fifteen guineas for every chain of the base. He desires to know what dimensions he must take, so as to have the most land for the least money; with the price it will cost him per acre. 3. Suppose a weight of 6 oz. is fixed to the end of an inflexible line, void of gravity, 40 inches long; required where another weight of 4 oz. must be fixed to the same line, so that the pendulum may vibrate quickest about the other end. 4.-Given the base and difference of the sides to determine the triangle, when the rectangle of the longest side and difference of the seg ments of the base is equal to the square of the shortest side. 5. From a given circle to cut off an arc, such that the rectangle under its sine, and the difference of its sine and cosine, may be a maximum. 6.-A staff, equal in length to the depth of a cylinder containing 48 ale gallons, being put diagonally to the bottom, rested with its upper end against the side, at one foot below the top. Quere, the cylinder's dimensions. 7.-A gentleman, when he first saw his present lady, was four times her age; twelve years after, on their wedding-day, he was only twice her age. 8. A gentleman sold a horse for 787, by which he gained half as much per cent. as the horse cost him. How much was that? 9.-On a division of the House of Commons, if the number for the motion had been increased by 40 from the other side, the question would have been carried by 3 to 2; but, if those against the motion had received 60 of the other party, the motion would have been defeated by 2 to 1. Query, did the motion succeed? and how many members were there in the house? 10.-Investigate a formula for approximating to the mth root of any number; and apply to finding the value of 161900 to 12 places of decimals? 11. When a parish was inclosed, the allotment of one of the proprietors consisted of two pieces of ground; one of which was in the form of a right-angled triangle; the other was a rectangle, one of the sides of which was equal to the hypothenuse of the triangle, the other to half the greater side: but, wishing to have his land in one piece, be exchanged his allotments for a square piece of ground of equal area, one side of which equalled the greater of the sides of the triangle which contained the right angle. By this exchange, he found that he had saved 10 poles of railing. What are the respective areas of the triangle and rectangle; and what is the length of each of their sides? 12. Two men set out at the same time, the one from London to Newcastle, and the other from Newcastle to London. He, from London, walked 10 miles the first day, 16 the second, 22 the third, and so on in arithmetical progression: he, from Newcastle, walked 20 miles every day. When will they meet? 1 13.-Required five numbers in geometrical progression, so that the sum of the two greatest may be 540, and the sum of the two least 20. 16. Of all those parabolic conoids which may be inscribed in tha cone whose altitude is a, and semi-base = b; query, that whose solidity is a maximum. 17.-In an oblique-angled plane triangle, there is given the difference of the sides which includes the angle of 71° 10′, equal to 11, and the line that bisects the said angle is equal to 24; from whence is required a theorem that will determine the base and sides of the said triangle. 18. Suppose a stick, perfectly cylindrical, 50 inches long, and length of the arm 20 inches; query, the centre of percussion, supposing a 2lbs. weight fastened 4 inches from the hand, and 5 lbs. quite at the other end. 19.-Let ABCD be a geometrical square, S a spring in an adjacent field, and let there be given the three distances SA=70, SB = 40, and SC60 chains; from thence to find the area and side of the said square. • 20.-A rider set out with a certain sum of money. At the first town he came to, he received 4807. He remitted the half of what he then had to his employer. At the second town he received 4004. and remitted half his cash; and at the third town he received 1201. and remitted one-third of his cash: he then had just the sum remaining with which he set out. How much was it? 21.-A, B, C, trade in company with a stock of 2004.; the sum advanced by A, the sum advanced by B, and the gain, being as 3, 5, and 4; and what C put in, when added to the gain, was as much as A and B put in together. What did each advance? 22. It is required to find the three sides of a right-angled triangle, from the following data:-The number of square feet in the area is equal to the number of feet in the hypothenuse + the sum in the other two sides; and the square described upon the hypotheneuse is less than the square described upon a line equal in length to the two sides, by half the product of the numbers representing the base and area. 23.-Given the base, the perpendicular, and the ratio, of the two sides of a triangle; to find the sides. 25-If the sum of the distances of the hours of three and four in the afternoon, from the meridian, or 12 o'clock line, be 82° 48′; query, the latitude for which this horizontal dial was made. 26.-The difference of longitudes of two places under the arctic circle being given 52°; to determine the sun's declination when he begins to appear to the inhabitants of one place, at the same instant of time that he sets to those of the other. 27.-The fences of a piece of land are the abscissa, ordinate, and curve of the conic parabola. Now there is given the perimeter or sum of the three sides = 80 chains, and the ordinate is to the abscissa as 6 to 5: required the area by a simple equation. 28. A gentleman bought a field in the form of a right-angled triangle, the sum of whose sides is 60 chains, at the following rate, viz. for every chain of the perpendicular he was to pay two guineas, and for each chain of the base one guinea. Query, the sides of the field where the gentleman has the most land in proportion to his money. 29.-Suppose a globe of wood, when put in water, be observed to rise two inches above the surface of the water, and afterwards, when put into another liquor whose specific gravity to that of water is as .92989 to 1, it rises only 1.25 inch above the surface of the liquor: it is required to determine the globe's diameter. 30.-What angle must a projectile make with the plane of the horizon, with a given velocity (a) per second, to describe in its flight the greatest area possible? 31.-A number of gentlemen at a tavern pay 61. If there had been two more, each person would have paid 5s. less: required the number of gentlemen, and the portion which each gentleman paid of the reckoning. 32.-To divide a given angle into two parts, such that their sines may have the given ratio of m to n. 33.-Required the value of x in the equation br-2 34.—Given the side of the base of a square pyramid = 45 inches, and its altitude 2 times the diameter of a circle whose area is equal to that of the base of the pyramid: at what distance from the vertex must the same be cut (by a plane parallel to the base), that the solidity of the lesser pyramid may be to the solidity of the frustrum in the ratio of 2 to 1? 35.-A gentleman, having a garden in the form of an ellipsis, whose transverse diameter is 300 and conjugate 210 feet, is determined to raise the said garden-plot one foot higher, by a trench that he will make round it. Required the breadth and depth of the trench, supposing them equal. 36.-Given the perpendicular let fall from the right angle of a rightangled triangle = 55; to determine the area of the triangle, when the same is a minimum. 37.—An erect declining dial declines from the south 30 degrees, and the plane's difference of longitude exceeds the substyle's distance from the meridian just equal to the co-latitude of the place. To determine in what latitude this dial is fixed. 38.-It is required to find two numbers, such that the product of the greater and square root of the less may be equal to 48, and the product of the less and square root of the greater may be 36. 39. In a mixture of rum and brandy, the difference between the quantities of each is, to the quantity of brandy, as 100 is to the number of gallons of rum; and the same difference is to the quantity of rum as 4 to the number of gallons of brandy. How many gallons are there of each? 40.-A market-woman, with a basket containing a certain number of eggs, sold half that number and half an egg to one person; half the remainder and half an egg to a second; half the remainder and half an egg to a third; and then had one egg left. How many had she at first? 41.-To find the area of a plane triangle, when two of its sides and the ¡ncluded angle are given. 42.-Reduce the surd to an equivalent; one having no deno minator affected by the radical sign. 43.-A person having doubled, at the gaming-table, the money which he had before he began to play, gave a sovereign to his domestic. Gaining, a second time, enough to double the money which he had remaining, he purchased, for a sovereign, a lottery-ticket, which came up a blank. Approaching the gambling-table for the third time, and doubling his money, he found that he had only a sovereign remaining in his pocket. Required what money he bad at first? 44.-Required a vulgar fraction equivalent to the repeating decimal 0.142857142857, &c. 45.-Given ' + y* = 130721 = a, x*y3 = 3961758400000 = b; query, and y. 46 -The diameters of an ellipsis are 60 and 40; what is the length and breadth of an oblong inscribed therein, whose content is 1152? 47.—In a right-angled plane triangular field, the legs are 3x* and *** ; and the line that bisects the right angle = x** chains: what is the content in acres? 48. Ye who can, by the pow'r of mystic lore, Their product just a lady's age will show. Given (w) × √(x3)=2x", √(2xw) = x×V (w). 49.-Required the area of the greatest right-angled triangle that can possibly be inscribed in an ellipsis, whose transverse and conjugate dia. meters are 70 and 50 inches respectively. 50.-A person paid a bill of 50%. with,half-guineas and crowns, using in all 101 pieces. How many pieces were there of each sort? 53. What quantity of canvas will be necessary for.forming a conical tent, whose height is 8 feet, and the diameter at bottom 13 feet? 54.-How many square feet of board are required to make a rectangular box, whose length is to be 34 feet, breadth 2 feet, and depth 20 inches? 55.-Suppose that a county contains one hundred thousand inhabi tants, and that the population increases every year by a thirtieth part ; what will be the number of inhabitants at the end of a century? 56.-A cornfactor bought wheat at 5s. per bushel, and barley at 2s. per bushel: now, in laying out 100 guineas, he observed that, if the square of the number of the bushels of wheat were multiplied by the number o. the bushels of barley, the product would be a maximum. How many bushels of each sort did he buy? 57.-A gentleman, by will, left legacies to his four servants, A, B, C, and D, proportional to the time each had been in his service; but, in case any of them should die before the expiration of the year, their share or shares to be divided equally among the others: accordingly, B died, and his share, so divided, made the share of C a mean proportional to those of A and D; whereas, before, A was to have 781., C 30l. and D 6. What money did each receive? 58. Given the difference of the two parallel sides of a trapezoid= 5.642224 poles, and the area one acre: query each side when the perimeter is a minimum. 59.-It is required how many times a conical glass, whose depth is 4 inches, and its internal concave surface is double the area of its base, or brim, may be filled out of a punch-bowl, whose form is a parabolic conoid, its depth 12 inches, and its internal concave surface double the area of its base, (as in the cone,) by a quadratic equation. 60.-Required the contents of the solid generated by the revolution of the cissoidal area AMP about the axis AB? (See Figure, p. 428.) 61.-Express the surd 3/9720 in its most simple form. 63 62.-The area of a right-angled triangle, whose sides are in arithmetical progression, being given equal to 216; to determine the triangle. 63.-Required two numbers, such that twice their sum, three times their product, and the difference of their square, shall be equal to one another. 64.-Given the solidity of the sector of a sphere = 1413.72, and the radius = 15; to determine the greatest inscribed cylinder. 65.-A water-mill is to be built where the current of water has a fall of 22 feet, or perpendicular descent: the architect desires to know the diameter of the water-wheel, so that the power or force it shall receive from the issuing water may be the greatest possible? 66.-Required the perimeter of the least common parabola that will circumscribe a circle whose diameter is 32 inches. 67. What numbers are those which, being both multiplied by 27, the first product is a square, and the second the root of that square; but, being both multiplied by 3, the first product is a cube, and the second the root of that cube? 68.—A farmer has two cubical stacks of hay; the side of one is 3 yards longer than the side of the other, and the difference of their contents is 117 solid yards. Required the side of each? 70.-Required the fluxion of x √ (a2 + x2)? 71.-What is the side of that equilateral triangle, whose area cost as much paving, at 8d. per foot, as the pallisading the three sides did at a guinea a yard? |